What are Statistics? |  Probability | Methods of Collecting, Representing, & Displaying Data | Data Calculations | California High School  Exit Exam References

Data Calculations
CA GR5 SDAP 1.3, CA GR6 SDAP 1.1 & 1.2 & 1.3 & 1.4 CA GR7 SDAP 1.3

Measures of Central Tendency (mean, median, mode)

A measure of central tendency is information that conveys something about the "middle-ness" of some data. It is information that takes in the "big picture" of how the data is organized and compresses that into one measurement that captures something about the essence of the "ordinary middle" of the data. The three measures of central tendency are mean, median, and mode. Each one tells a slightly different story about the middle because each is figured out differently.

The mean measures the middle that would happen if all shared equally. So if 5 guys have $20, $100, $10, $5, and $10, the mean would be calculated by pooling the money (add all the data elements together) and splitting it back up (divide by the number of data elements). 

In this example,

mean = $(100+20+10+10+5)/5 = 145/5 = $49. 

The median measures the middle that you get from lining them up in order and counting to the center. So we line the 5 guys up as 100, 20, 10, 10, 5; the center is the 3rd entry, so median = $10. If we had 6 entries, the middle would be between the 3rd and 4th entry, so just average the two to get the median. 

The mode measures the most common result. Notice that $10 showed up twice while no other amount did, so mode = $10. If there are competing modes, for example if $5 had also appeared twice, then just list them all (multiple modes are allowed, bimodal means two modes exist, trimodal means three modes exist, and so on).

Example 4

Consider the data {11.5, 23.6, 18.2, 19.0, 15.5, 16.5, 18.0, 12.2, 13.9, 19.4). Even without knowing what the data represents and how it was collected, you can compute the mean, median, and mode. To calculate the mean, sum up the ten numbers and divide by 10, getting 16.78. To get the median, put them in order and find the middle. Since there are 10 entries, the median has to have 5 higher than it and 5 lower, so the median is between the 5th and 6th entries, which are 16.5 and 18. The proper procedure is to take the average of those two entries, so the median is 17.25. There is no mode for the data because each entry is unique an no one is any more common than the other. 

Example 5

Find the error if a teacher says that for the mean is 85, the median is 100, and the mode is 80 when the quiz scores were 80, 75, 90, 70, 100, 75, 100, 75, 90, 80, 100, 70, 75, 100, 80, and 100. 
The median can't be 100 - 100 is the highest score, not the middle score. However, 100 occurs five times, more often than any other score, so 100 is the mode. The teacher probably figured out the right numbers but then switched the median with the mode. The teacher should always remember that the median is the middle that has the same length of the list of items above it and below it, just like the median line on a road has the same width or road to the left as it does to the right. 

Measures of Variability (range, variance, standard deviation)

Measures of variability are the opposite of measures of central tendency. Instead of gathering the essence of the middle of the data, they gather the essence of how widely spread the data is. The simplest measure of variability is the range, which is simply the distance between the ends of the data. It is the width from smallest to largest data element. Just subtract the smallest value from the largest. So for the 5 guys in the explanation, you get a range of 95, because the most amount of money, $100, has the least amount, $5, subtracted from it. There's quite some disparity in the amount of money each has, and the largeness of $95 with respect to most of the entries shows this. If another group of several people had a range of only $8 in the amount of money they had, you'd know the cash was distributed much more equally, that is, with less variation or variability. In Example 4, the range is 12.1, which comes from 23.6 - 11.5. In Example 5 the range is 30, from maximum 100 subtract minimum 70.